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Laplace Transform
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
In this section, we will not use Eq. (5.4.1), as it is very complicated. Instead, we consider three practical methods to find the inverse Laplace transform: Partial Fraction Decomposition, the Convolution Theorem, and the Residue Method. The reader is free to use any of them or all of them. We will restrict ourselves to finding the inverse Laplace transform of rational functions or their products on exponentials; that is, F(λ)=P(λ)Q(λ)orFα(λ)=P(λ)Q(λ)e−αλ,
Laplace Transforms
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
In this section, we will not use Eq. (5.4.1, as it is very complicated. Instead, we consider three practical methods to find the inverse Laplace transform: Partial Fraction Decomposition, the Convolution Theorem, and the Residue Method. The reader is free to use any of them or all of them. We will restrict ourselves to finding the inverse Laplace transform of rational functions or their products on exponentials; that is, F(λ)=P(λ)Q(λ)orFα(λ)=P(λ)Q(λ)e-αλ, $$ F(\lambda ) = \frac{{P(\lambda )}}{{Q(\lambda )}}\,\,{\text{or}}\,\,F_{\alpha } (\lambda ) = \frac{{P(\lambda )}}{{Q(\lambda )}}e^{{ - \alpha \lambda }}, $$
Combinatorics
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
In Example 1.10 we used the partial fraction decomposition to break apart a rational function to find an exact formula for the coefficients. This idea can be generalized, allowing us to find exact (or asymptotic) formulas for the coefficients of any rational function, which is expressible as a power series about z=0.
Analytical prediction of stress and strain in adhesive tube-to-tube joints under thermal expansion/contraction
Published in The Journal of Adhesion, 2023
Haotian Liu, Justin A. Weibel, Eckhard A. Groll
The original study by Lubkin and Reissner[1] provides no details on the solution procedure beyond a remark that the equation set is linear with constant coefficients and thus can be solved with standard methods. Nevertheless, in the pursuit of an explicit closed-form solution, the high order of the set of differential equations is a major practical difficulty. While it is possible to achieve approximate solutions without particular difficulty by numerical integration, we seek an analytical solution as discussed in the introduction. Goglio and Paolino[23] revisit the Lubkin and Reissner[1] model and provide a solution method using Laplace transforms. A similar solution approach using Laplace transforms is applied in this study, but the procedure of finding the partial fraction decomposition is modified to simplify the calculation. The solution process is evaluated using Mathematica. The algorithm used to solve the set of governing equations is shown in Figure 3, where f(z), g(z), and h(z) are three dimensionless functions and is a dimensionless variable such that at the left end of the adhesive and at the right end of the adhesive. The detailed expressions of the dimensionless functions and their functions can be found in the Supplementary Material section S2.
Exponential Time Differencing Schemes for Fuel Depletion and Transport in Molten Salt Reactors: Theory and Implementation
Published in Nuclear Science and Engineering, 2022
Zack Taylor, Benjamin S. Collins, G. Ivan Maldonado
While the Padé approximation is indeed a rational function approximation, it differs from the methods presented in this section. The rational functions presented here are represented in partial fraction decomposition form. These types of methods are algorithms that were developed by transforming the matrix exponential into the complex plane using Cauchy’s integral formula: